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### BENFORD’S LAW

SAS 99: Detecting Fraud Using Benford’s Law

FAE/NYSSCPA

TECHNOLOGY ASSURANCE COMMITTEE

March 13, 2003

Christopher J. Rosetti, CPA, CFE, DABFA

KPMG

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Benford’s Law

Number Log Formula

10 1.0000 LOG(10)

11 1.0414 LOG(11)

12 1.0792 LOG(12)

13 1.1139 LOG(13)

14 1.1461 LOG(14)

15 1.1761 LOG(15)

16 1.2041 LOG(16)

17 1.2304 LOG(17)

18 1.2553 LOG(18)

19 1.2788 LOG(19)

20 1.3010 LOG(20)

Logarithm Example

Multiply 320 by 417(Answer 133,440)

Log(320) =2.50515

Log(417) =2.620136

Log (320) + Log (417) =5.125286

10^5.5125286 =133,440

Note on the Frequency of Use of the Different Digits in Natural Numbers

Theory: “A multi-digit number is more likely to begin with ‘1’ than any other number.” In other words, these are probably the most faded numbers on our calculators.

Newcomb’s Shortcoming

He failed to provide a reason why his theory and formula worked!!!!!

Frank Benford

Noted the same phenomena as Newcomb in the same exact manner in the late 1920’s, and theorized that unless his friends had a predilection for low digit numbers, there must be a reason to explain this phenomena.

Benford Tests

Analyzed 20,229 sets of numbers, including, areas of rivers, baseball averages, numbers in magazine articles, atomic weights of atoms, electricity bills on the Solomon Islands, etc.

Benford’s Conclusion

- Multi digit numbers beginning with 1, 2 or 3 appear more frequently than multi digit numbers beginning with 4, 5, 6, etc.
- The frequency of which these digits appear in nature was published in “The Law of Anomalous Numbers”

Percentages

Percentages

Digit - Position in Number

- 1st 2nd 3rd1. 301 .113 .1013
- 2. 176 .108 .1009
- . 124 .104 .1005
- . 096 .100 .1001
- . 079 .096 .0997
- 6 .066 .093 .0994

Percentages

First Digit First Digit First Digit

1 2 3

Area Rivers 31 16.4 10.7

Populations 33.9 20.4 14.2

Newpapers 30 18 12

Pressure 29.6 18.3 12.8

Mol. Weight 26.7 25.2 15.4

Atomic Weight 47.2 18.7 5.5

X-Ray Volts 27.9 17.5 14.4

Batting Averages 32.7 17.6 12.6

Death Rate 27 18.6 15.7

Average 30.6 18.5 12.4

Probable Error 0.8 0.4 0.4

Conclusion Cont.

- The number 1 predominates every step of most progressions.
- Stock Market example: Assume 20% annual return on a $1,000 investment. It takes 4 years for the stock to go from $1,000 to $2,000, approximately 3 years to go from $2,000 to $3,000, approximately 2 years to go from $3,000 to $4,000. Before long you start over at 1 or $10,000.

Conclusion Cont.

Months in which Investment ranged between:

$1,000 and $1,999 41 29.50%

$2,000 and $2,999 25 17.99%

$3,000 and $3,999 17 12.23%

$4,000 and $4,999 14 10.07%

$5,000 and $5,999 11 7.91%

$6,000 and $6,999 9 6.47%

$7,000 and $7,999 8 5.76%

$8,000 and $8,999 7 5.04%

$9,000 and $9,999 7 5.04%

Stock Market Example

- Sample of 12,00 stock market quotes from the Wall Street Journal.

Stock Market Example

Actual Expected Actual Expected

Frequency Frequency Frequency Frequency Difference

Digit 1 3364 3619 27.98% 30.10% -2.12%

Digit 2 1554 2116 12.93% 17.60% -4.67%

Digit 3 1182 1502 9.83% 12.49% -2.66%

Digit 4 1240 1165 10.31% 9.69% 0.62%

Digit 5 1026 952 8.53% 7.92% 0.61%

Digit 6 1103 804 9.17% 6.69% 2.48%

Digit 7 897 697 7.46% 5.80% 1.66%

Digit 8 820 616 6.82% 5.12% 1.70%

Digit 9 836 551 6.95% 4.58% 2.37%

12,022

Newcomb vs. Benford

- Benford also did not have an explanation for this phenomena, however, at least he had evidence that demonstrated the laws ubiquity.
- The theory remained unchallenged, but failed to generate any publicity.

1961

- Research conducted revealed that Benford’s probabilities are scale invariant, therefore, it doesn't’t matter if the numbers are denominated in dollars, yens, marks, pesos, rubbles, etc.

Betting

- Other than proving the financial reasonableness of forecasts, the main use for Benford’s Law was used for making money by betting with unsuspecting friends.

Mark Nigrini

- In 1992, Nigrini published a thesis noting that Benford’s Law could be used to detect fraud.

How Does this help us?

- Because human choices are not random, invented numbers are unlikely to follow Benford’s Law, I.e., when people invent numbers, their digit patterns (which have been artificially added to a list of true numbers) will cause the data set to appear unnatural.

Source: Mark Nigrini

Five Major Digit Tests.

- 1st digit test
- 2nd digit test
- First two digits
- First three digits
- Last two digits

Source: Mark Nigrini

First Digit Test

- High Level Test
- Will only identify the blinding glimpse of the obvious
- Should not be used to select audit samples, as the sample size will be too large.

Sourec: Mark Nigrini

Second Digit Test

- Also a high level test
- Used to identify conformity
- Should not be used to select audit samples

Source: Mark Nigrini

First Two Digits Test

- More focused
- Identifies manifested deviations for further review
- Can be used to select audit targets for preliminary review

Source: Mark Nigrini

First Three Digits Test

- Highly Focused
- Used to select audit samples
- Tends to identify number duplication

Source: Mark Nigrini

Last Two Digits Test

- Used to identify Invented (overused) and rounded numbers
- Expected proportion of all possible last two digit combinations is .01

Source: Mark Nigrini

Not all Data Conforms!!!!!!!!!

- The data set should describe similar data (populations of towns)
- Artificial limits should not exist (no minimum sale amount)
- The data can’t consist or pre-arranged numbers (SSN, Tel Numbers)
- The data should consist of more small items than large items

Not all Data Conforms

- The data should not be a subset of a set
- Does not work if data has been aggregated, I.e. daily deposits are combined and recorded weekly
- Data should relate to s specific period
- The data population should be large enough so that the proportions can manifest themselves

Fraud Cases

- What will you generally see:
- Fraudster starts out small then increases the dollar amount. The amounts will be just below a limit that requires further review. The numbers will not follow a digital pattern. The amounts will not be rounded, and certain digit patterns will be repeated.

Source: Mark Nigrini

Example

- Examined over 1,000 cash disbursements (entire population) during the year (amounts over $500 required 2 signatures and amounts over $5,000 required competitive bids).
- Sample is on next slide

Example

Amount Description Check. No.

$225.95 SEIU - LU 82 ED ASSES FUND 6/98. 4001

$1,212.97 SCHINDLER ELEV CORP JUN 98. 4002

$4,999.50 YORK INT CORP - 7/98-9/98. 4003

$339.13 US FOODSERVICE 10/29/98. 4004

$473.98 VIRGINIA DEPT OF TAXATION JUNE '98 4005

$250.81 W W GRAINGER INC - SUPPLIES 4006

$504.00 LJC LIGHTING SUPPLY - LIGHT BULBS. 4007

$171.70 CLERK, DC SUPERIOR COURT 12/25/98. 4008

$225.15 SEIU - SEIU LU 82 ED ASSES FD 9/98. 4009

$477.26 VIRGINIA DEPT OF TAXATION -1998. 4010

Actual First and Second Digit Frequency

Regular payroll garnishment..

Kay Grogan Food/Bev. Company uses ARAMARK.

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Monthly supply contract for $303.

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Possible structuring to avoid authorization thresholds.

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Pest Control.

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Maint. Contract.

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Applying Benford’s Law

- Income tax agencies.
- Audits of Accounts Payable (I/A, Ext. Auditors, Fraud Examiners, etc).
- Expenses reimbursements.

Who Uses This

- US West, Sprint, Colgate, P&G, Nortel, American Airlines, United Airlines, Ameritech, Lockheed Martin, KPMG, ARCO, State of Texas.

Source: Mark Nigrini

Cost of Data Analysis Software

- $245 for 13 programs which run on Excel 97 or Excel 2000.
- $795 for all programs. Works with ACL and Idea.

Source: Mark Nigrini

Caution

- Does not work with Lottery
- May not work for certain types of expenses in which documentation is not required for expenses under a certain category.
- Authorization Levels.

Caution

CAUTION

- It only works with natural numbers (those numbers that are not ordered in a particular numbering scheme, I.e., telephone numbers, social security numbers.

Caution

CAUTION

- The sample should be large enough so that the predicted proportions can assert themselves, and they should be free of artificial limits. I.E., don’t analyze the prices of 10 different types of beer, as the sample is small and the prices are forced by competition to stay within a narrow range.

Summary

STOP

- Benford’s Law provides a data analysis method that can help alert us to possible errors, biases, potential fraud, costly processing inefficiencies or other irregularities.

Articles

STOP

- Journal of Accountancy (5/99)
- New Scientist (7/99)
- Internal Auditor (2/99)
- Inside Fraud Bulletin (3/99)
- Auditing: A Journal of Practice & Theory (Fall of 1997)

Articles Continued

STOP

- White Paper (4/94)
- White Paper (9/99)
- New York Times (8/4/98)
- Information Technology (9/97)

Christopher J. Rosetti, CPA, CFE, DABFA, is a Director for KPMG's Forensics unit. He previously spent four years conducting financial statement audits for KPMG before leaving to work as a confidential investigator for a governmental investigative agency. In addition to conducting fraud investigations and compliance and integrity reviews for KPMG, Chris spent five years teaching accounting related courses at Siena College and developed and taught a forensic accounting course for Utica College. He has authored seven articles, qualified and testified as an expert witness and given more than 50 fraud and/or forensic presentations to various private and public organizations. Chris also served also a National Fraud Instructor for KPMG's recruits, has been a presenter at the last three annual seminars hosted by the Association of Certified Fraud Examiners (ACFE), has developed training and self study courses for the ACFE, and is an instructor for the Person Wolinsky CPA Review Course.

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